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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8216, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,0,52,2]))
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pari:[g,chi] = znchar(Mod(9,8216))
\(\chi_{8216}(9,\cdot)\)
\(\chi_{8216}(81,\cdot)\)
\(\chi_{8216}(321,\cdot)\)
\(\chi_{8216}(737,\cdot)\)
\(\chi_{8216}(913,\cdot)\)
\(\chi_{8216}(1257,\cdot)\)
\(\chi_{8216}(1537,\cdot)\)
\(\chi_{8216}(2401,\cdot)\)
\(\chi_{8216}(2577,\cdot)\)
\(\chi_{8216}(2785,\cdot)\)
\(\chi_{8216}(2889,\cdot)\)
\(\chi_{8216}(3097,\cdot)\)
\(\chi_{8216}(3233,\cdot)\)
\(\chi_{8216}(3337,\cdot)\)
\(\chi_{8216}(3753,\cdot)\)
\(\chi_{8216}(3961,\cdot)\)
\(\chi_{8216}(4377,\cdot)\)
\(\chi_{8216}(4449,\cdot)\)
\(\chi_{8216}(5177,\cdot)\)
\(\chi_{8216}(5385,\cdot)\)
\(\chi_{8216}(6529,\cdot)\)
\(\chi_{8216}(6561,\cdot)\)
\(\chi_{8216}(6633,\cdot)\)
\(\chi_{8216}(7081,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((2055,4109,3161,3953)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{1}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 8216 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
